Abstract
We show that the Föppl-von Kármán theory arises as a low energy Γ-limit of three-dimensional nonlinear elasticity. A key ingredient in the proof is a generalization to higher derivatives of our rigidity result [5] that for maps v : (0, 1)3 → ℝ3, the L2 distance of ∇v from a single rotation is bounded by a multiple of the L2 distance from the set SO(3) of all rotations.
| Translated title of the contribution | The Föppl-von Kármán plate theory as a low energy Γ-limit of nonlinear elasticity |
|---|---|
| Original language | French |
| Pages (from-to) | 201-206 |
| Number of pages | 6 |
| Journal | Comptes Rendus Mathematique |
| Volume | 335 |
| Issue number | 2 |
| DOIs | |
| State | Published - Jul 15 2002 |
Bibliographical note
Funding Information:Acknowledgement. RDJ thanks AFOSR/MURI (F49620-98-1-0433) and NSF (DMS-0074043) for supporting his work. GF and SM were partially supported by the TMR network FMRX-CT98-0229. References [1] S.S. Antman, Nonlinear Problems of Elasticity, Springer, New York, 1995. [2] G. Anzelotti, S. Baldo, D. Percivale, Dimension reduction in variational problems, asymptotic development in Γ-convergence and thin structures in elasticity, Asymptotic Anal. 9 (1994) 61–100. [3] P.G. Ciarlet, Mathematical Elasticity II – Theory of Plates, Elsevier, Amsterdam, 1997. [4] D.D. Fox, A. Raoult, J.C. Simo, A justification of nonlinear properly invariant plate theories, Arch. Rational Mech. Anal. 124 (1993) 157–199. [5] G. Friesecke, R.D. James, S. Müller, Rigorous derivation of nonlinear plate theory and geometric rigidity, C. R. Acad. Sci. Paris, Série I 334 (2002) 173–178. [6] G. Friesecke, R.D. James, S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., to appear. [7] H. LeDret, A. Raoult, Le modéle de membrane non linéaire comme limite variationelle de l’élasticité non linéaire tridimensionelle, C. R. Acad. Sci. Paris, Série I 317 (1993) 221–226. [8] H. LeDret, A. Raoult, The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. 73 (1995) 549–578. [9] A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th edn., Cambridge University Press, Cambridge, 1927. [10] J.J. Marigo, H. Ghidouche, Z. Sedkaoui, Des poutres flexibles aux fils extensibles : une hiérachie de modèles asymptotiques, C. R. Acad. Sci. Paris, Série IIb 326 (1998) 79–84. [11] R. Monneau, Justification of nonlinear Kirchhoff–Love theory of plates as the application of a new singular inverse method, Preprint, 2001. [12] O. Pantz, Une justification partielle du modèle de plaque en flexion par Γ-convergence, C. R. Acad. Sci. Paris, Série I 332 (2001) 587–592. [13] A. Raoult, Personal communication.